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Homemodel

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# model

Let $\tau$ be a signature and $\varphi$ be a sentence over $\tau$. A structure $\mathcal{M}$ for $\tau$ is called a *model* of $\varphi$ if

$\mathcal{M}\models\varphi,$ |

where $\models$ is the satisfaction relation. When $\mathcal{M}\models\varphi$, we says that $\varphi$ *satisfies* $\mathcal{M}$, or that $\mathcal{M}$ is *satisfied by* $\varphi$.

More generally, we say that a $\tau$-structure $\mathcal{M}$ is a *model* of a theory $T$ over $\tau$, if $\mathcal{M}\models\varphi$ for every $\varphi\in T$. When $\mathcal{M}$ is a model of $T$, we say that $T$ *satisfies* $\mathcal{M}$, or that $\mathcal{M}$ is satisfied by $T$, and is written

$\mathcal{M}\models T.$ |

Example. Let $\tau=\{\cdot\}$, where $\cdot$ is a binary operation symbol. Let $x,y,z$ be variables and

$T=\{\forall x\forall y\forall z\left((x\cdot y)\cdot z=x\cdot(y\cdot z)\right)\}.$ |

Then it is easy to see that any model of $T$ is a semigroup, and vice versa.

Next, let $\tau^{{\prime}}=\tau\cup\{e\}$, where $e$ is a constant symbol, and

$T^{{\prime}}=T\cup\{\forall x(x\cdot e=x),\forall x\exists y(x\cdot y=e)\}.$ |

Then $G$ is a model of $T^{{\prime}}$ iff $G$ is a group. Clearly any group is a model of $T^{{\prime}}$. To see the converse, let $G$ be a model of $T^{{\prime}}$ and let $1\in G$ be the interpretation of $e\in\tau^{{\prime}}$ and $\cdot:G\times G\to G$ be the interpretation of $\cdot\in\tau^{{\prime}}$. Let us write $xy$ for the product $x\cdot y$. For any $x\in G$, let $y\in G$ such that $xy=1$ and $z\in G$ such that $yz=1$. Then $1z=(xy)z=x(yz)=x1=x$, so that $1x=1(1z)=(1\cdot 1)z=1z=x$. This shows that $1$ is the identity of $G$ with respect to $\cdot$. In particular, $x=1z=z$, which implies $1=yz=yx$, or that $y$ is a inverse of $x$ with respect to $\cdot$.

Remark. Let $T$ be a theory. A class of $\tau$-structures is said to be *axiomatized by* $T$ if it is the class of all models of $T$. $T$ is said to be the *set of axioms* for this class. This class is necessarily unique, and is denoted by $\operatorname{Mod}(T)$. When $T$ consists of a single sentence $\varphi$, we write $\operatorname{Mod}(\varphi)$.

## Mathematics Subject Classification

03C95*no label found*

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